Todd W Neller Dept of Computer Science Dudo According to legend King Atahualpa of the Incas taught this to the Spanish conquistador Pizarro more than 400 years ago Mohr 1997 Origin generally believed to be 15 ID: 226497
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Slide1
Dudo: An Introduction to the Incan Bluffing Dice Game
Todd W. Neller
Dept. of Computer ScienceSlide2
Dudo
“According to legend, King Atahualpa of the Incas taught this to the Spanish conquistador Pizarro more than 400 years ago…” (Mohr, 1997)
Origin generally believed to be 15
th
c. Inca
Variants long popular in many Latin American countriesSlide3
Beyond Latin America
Many variants/names (Liar’s Dice, Perudo, Bluff, Call My Bluff, Cacho, Cachito)
Internationally popular
BoardGameGeek.com rank 236/49160 (top ½%!), 11/2/10Slide4
Dudo Overview
Bluffing dice game for 2+ players
Each player rolls 5 dice concealed under cup
Players make successively bolder claims about
all
dice rolled until player challenges
Loser of challenge loses dice
Last player with dice winsSlide5
Dudo ClaimsSlide6
Dudo Rules
Players each roll and privately view 5 dice concealed under a cup. 1’s are wild.
Players make successively greater claims until one challenges the previous claim with “Dudo!” (Sp. “I doubt it!”), all reveal dice, and:
More/less than claimed? Challenger/claimant loses dice according to difference.
Claim exactly correct? Challenger loses 1 die.
The next round begins with challenge winner.
The last player with dice wins.Slide7
Computer Play
Gather
around lab computers. (Carefully fold down desks.)
Commands:
cp ~
tneller
/pub/java/
dudo
/*.java
.
j
avac
*.java
java
DudoGame
Input claim format: “
dudo
” or “<num> <rank>” (e.g. “7 6”)
What do you observe?Slide8
Simple Dudo Player
SimpleDudoPlayer
follows a simple policy based on roll probabilities.
Make the strongest legal claim that is correct with probability ≥ ½.
If no such claim exists, call “
dudo
”. Slide9
Calculating Claim Probabilities
Example: What is the probability of exactly 2
of 5 dice being 6’s?
1’s are wild, so the probability of a single die being a 6 is 1/3.
6
5
different 5 dice rolls (in sequence)
5 choose 2 = 5! / (2! 3!) = 120 / (2 * 6) = 10 different ways of choosing 2 from 5
Each occurrence has prob. (1/3)
2
(1-1/3)
3
Answer:
10 ×
(1/3)
2
(1-1/3)
3
= 80/243Slide10
Calculating Claim Probabilities (cont.)
Example: What is the probability of
at least
2
of 5 dice being 6’s?
Exactly
0: 1
×
(1/3)
0
(1-1/3)
5
= 32/243
Exactly
1: 5
×
(1/3)
1
(1-1/3)
4
= 80/243
Exactly
2:
10 ×
(1/3)
2
(1-1/3)
3
= 80/243
Exactly 3: 10
×
(1/3)
3
(1-1/3)
2
= 40/243
Exactly
4: 5
×
(1/3)
4
(1-1/3)
1
= 10/243
Exactly 5:
1
×
(1/3)
5
(1-1/3)
0
= 1/243
At
least 2: (
80 + 40 + 10 +
1) / 243 = 131/243 ≈ .539 Slide11
Choosing a Claim
Example computation:
SimpleDudoPlayer
chooses the strongest claim
that is correct with probability ≥ ½:
five 6’sSlide12
Game Information
Dudo is a game of information:
State information: What you hold, how many dice others have
Probabilistic information: Roll probabilities
“Tell” information: Behavior indicators
Play information: Other player actions
Now that you know how
SimpleDudoPlayer
makes decisions, exploit that information.Slide13
Mixed Strategy
Predictability
Exploitability
Are these also examples of games with optimal mixed strategy?
Tic
Tac
Toe
Rock Scissors Paper
Blackjack / Pig
Poker
Clue
Game-theoretic techniques help us
compute or
approximate optimal mixed strategiesSlide14
Conclusion
Dudo is an excellent, ancient bluffing dice game.
Simple
application of probability and combinatorial mathematics yields a reasonable simple player
.
Predictability
Exploitability
Because player choices yield information, games of information often call for mixed strategies.Slide15
References
Reiner
Knizia
.
Dice Games Properly Explained
. Elliot Right-Way Books, Brighton Road, Lower
Kingswood
,
Tadworth
, Surrey, KT20 6TD UK, 1999
Merilyn
Simonds
Mohr.
The New Games Treasury
– More Than 500 Indoor and Outdoor Favorites with Strategies, Rules and Traditions,
Houghton Mifflin Co., Boston, 1997
Dudo rules online:
http://cs.gettysburg.edu/~tneller/games/rules/dudo.pdf